Mixed-integer Linear Programming Models in Supply-Chain and Reliability Management

This project focusses on bilevel optimisation. Bilevel optimisation problems consist of two problems, defined as the leaders and the follower's problems and we can think of these two problems being solved sequentially. The leader shall go first, defining their variables such that they will maximise their objective. The follower, will then optimise their problem, with the leader's variables acting as parameters. One can think of this as the follower reacting to the leader. The follower's variables are contained within the leaders problem and will affect their objective. Hence, when the leader is optimising, they must ensure that the followers variables in their problem, will match the reaction of the follower, otherwise their solution is not a feasible one. Due to this reaction structure, bilevel optimisation has applications within, among others, the energy and transportation sectors. One of the state of the art techniques used to solve bilevel problems is to exchange the follower's problem for constraints which are necessary for optimality. One such method is known as the Value Function Approach, ensuring that the follower's variables in the leader's problem are indeed optimal for the follower.
However, the value function is a complex function, difficult to evaluate, which cannot always be explicitly included in the leader's problem. The aim of this project is to utilise the Value Function Approach for a specific bilevel problem in which the leader's variables affect the follower's objective, but not their variables.
The goal of this research is to conceive an efficient algorithm based on the Value Function Approach for this problem type, which shall be quicker than generic bilevel solvers applied to such problems.